Fast quantum state discrimination with nonlinear PTP channels

Michael R. Geller

Center for Simulational Physics, University of Georgia, Athens, Georgia 30602, USA

(Dated: November 10, 2021)

We investigate models of nonlinear quantum computation based on determinis-tic positive trace-preserving (PTP) channels and associated master equations. Themodels are defined in any finite Hilbert space, but the main results are for dimen-sion N = 2. For every normalizable linear or nonlinear positive map φ on boundedlinear operators X, there is an associated normalized PTP channel φ(X)/tr[φ(X)].Normalized PTP channels include unitary mean field theories, such as the Gross-Pitaevskii equation for interacting bosons, as well as models of linear and nonlineardissipation. They classify into 4 types, yielding 3 distinct forms of nonlinearity whosecomputational power we explore. In the qubit case these channels support Bloch balltorsion and other distortions studied previously, where it has been shown that suchnonlinearity can be used to increase the separation between a pair of close qubitstates, resulting in an exponential speedup for state discrimination. Building on thisidea, we argue that this operation can be made robust to noise by using dissipa-tion to induce a bifurcation to a phase where a pair of stable fixed points create anintrinisically fault-tolerant nonlinear state discriminator.

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Quantum nonlinearity, beyond the stochastic nonlinearity already provided by projec-

tive measurement, might be a powerful computational resource [1–14]. However there is

no experimental evidence for physics beyond standard linear quantum mechanics [15–18].

It is challenging to even formulate a consistent, fundamentally nonlinear quantum theory

in accordance with general principles [19–41]. In this paper we consider the application of

effective quantum nonlinearity to information processing, while at the same time accepting

that quantum physics is fundamentally linear. Effective means that it arises in some ap-

proximate (e.g., low-energy) quantum description, is a consequence of constraints on a linear

system, or emerges in the limit of a large number of particles [42–47]. An early example

from particle physics is the low-energy reduction of electroweak theory to Fermi’s simpler

1933 model L = ψi/∂ψ − g (ψγaψ)2, containing a four-fermion interaction. Effective nonlin-

earity is also common in condensed matter (Breuer and Petruccione [47] give an excellent

introduction). It’s a natural byproduct of dimensional reduction, where the dynamics of a

complex quantum many-body system is described by a model with fewer degrees of freedom.

A familiar example is self-consistent mean field theory: For n quantum particles moving in

D dimensions, such an effective model reduces a problem with nD degrees of freedom to

one involving only D, at the expense of nonlinearity and errors (usually). Examples include

mean field models for superfluids, superconductors, and laser fields. Beyond mean field the-

ory, various forms of nonlinearity have been proposed to describe friction and dissipation in

quantum mechanics [42, 45–51] and for open systems more generally [42, 45–53].

It is not known whether effective nonlinearity can actually be used to enhance quantum

information processing. Perhaps any nonlinear advantage is an artifact of approximations

and could never be realized [11–13]. Childs and Young [13] used the speedup predicted for

optimal qubit state discrimination with Gross-Pitaevskii nonlinearity to derive a complexity

theoretic bound on the long-time accuracy of the Gross-Pitaevskii equation itself (though

weaker than a known bound [13]). Can a different nonlinearity be used, or can the Gross-

Pitaevskii nonlinearity be used in another way? Is speedup the only possible advantage;

what about noise resilience? The purpose of this paper is to explore these questions by

providing a framework for studying known nonlinear channels from an information processing

perspective, and for proposing new ones that might be experimentally realizable in the

near future. The framework should be applicable to specially designed strongly correlated

quantum materials that are open and possibly operated under extreme conditions (think non-

3

Hermitian exciton-polariton condensates [54, 55]). We aim to impose as little structure as

possible on the allowed evolution beyond preserving the Hermitian symmetry, positive semi-

definiteness, and trace of the density matrix X. Stochastic nonlinearity [22, 27, 49], including

projective measurement with post-selection [56–59], is also a useful resource, but will not

be covered here. Therefore we restrict ourselves to deterministic nonlinear positive trace-

preserving (PTP) channels [60]. We do not attempt to derive an effective nonlinear theory

from an underlying physical model, but instead try to identify what types of nonlinearity

are desirable, and why. However we will only scratch the surface of this interesting but

formidable problem.

Interestingly, according to Abrams and Lloyd [2], Aaronson [7], and Childs and Young

[13], nonlinearity is not required in large quantities: Even one “nonlinear qubit” would

provide a computational benefit when coupled to a linear quantum computer. This is because

nonlinearity can be used to increase the trace distance between a pair of qubit states [1–

3, 7, 13], resulting in an exponential speedup for unstructured search and hence for any

problem in the class NP. We mainly focus on this N = 2 case. While these results are

intriguing, it must be emphasized that they apply in an idealized setting, where model

errors are neglected.

Building on a growing body of similarly motivated work [42–53], we study a family of

normalized PTP channels of the form φ(X)/tr[φ(X)], where φ(X) is a positive linear or

nonlinear map on bounded linear operators X satisfying tr[φ(X)] 6= 0. Normalized PTP

channels fall into 4 classes, yielding 3 distinct forms of nonlinearity. As in the classical

setting, rich dynamical structures result from the interplay of nonlinearity and dissipation,

and we will see that PTP channels allow for greater control over engineered linear dissipation

than completely positive channels do. Our main result is the identification of a nonlinear

channel where the Bloch ball separates into two basins of attraction, which can be used to

implement fast intrinsically fault-tolerant state discrimination. Although we do not address

the model error issue directly, we hope that the predicted phase will survive in realistic

models and be observable experimentally. Section I mainly covers the definition of a PTP

channel and can be skipped by most readers. Normalized PTP channels are classified in

Sec. II, and fault-tolerant nonlinear state discrimination is explained in Sec. III. Section IV

contains the conclusions.

4

I. PTP CHANNELS

A. Notation

Let H = (span{|ei〉}Ni=1, 〈x|y〉) be the system Hilbert space with inner product 〈x|y〉 =∑Ni=1 x

∗i yi, complete orthonormal basis {|ei〉}Ni=1 (〈ei|ej〉= δij,

∑Ni=1 eii = IN , eij := |ei〉〈ej|),

and norm ‖|x〉‖ =√〈x|x〉. Here x∗ denotes complex conjugation and IN is the N ×N

identity. Let X : H → H be a linear operator on H (isomorphic to a matrix X ∈ CN×N) and

let X† be its adjoint. The set of these bounded linear operators form a second complex vector

space B(H,C); this space is our main focus. Let Her(H,C) = {X ∈ B(H,C) : X = X†}

be the subset of self-adjoint observables, and Her≥0(H,C) = {X ∈ Her(H) : X � 0} be the

positive semidefinite (PSD) subset. Quantum states live in the subset of Her≥0(H,C) with

unit trace: Her≥01 (H,C) = {X ∈ Her≥0(H,C) : tr(X) = 1}. In the qubit case the elements

X = (I2 + r ·σ)/2 of Her≥01 (H,C) are mapped, using the basis of Pauli matrices (σa)a=1,2,3,

to real vectors r = (x, y, z) ∈ R3 with |r| ≤ 1, the closed Bloch ball B1[0].

B. Nonlinear positive maps

Definition 1 (Linear map). Let L : B(H,C) → B(H,C) be a map on bounded linear

operators satisfying (i) L(X + Y ) = L(X) + L(Y ), and (ii) L(αX) = αL(X), for every

X, Y ∈ B(H,C) and α ∈ C. Then L is a linear map on B(H,C).

Lemma 1. Let L : B(H,C) → B(H,C) be a linear map on finite-dimensional bounded

linear operators. Then L has a representation

X 7→ L(X) =m∑α=1

AαXBα, Aα, Bα ∈ CN×N , m≤N2, N = dim(H). (1)

Proof: Every linear map is specified by its action on a complete matrix basis eab =

|ea〉〈eb| ∈ CN×N , (eab)a′b′ = δaa′δbb′ :

X 7→ L(X) = L

( N∑a,b=1

Xab eab

)=

N∑a,b=1

Xab L(eab). (2)

5

The set {L(eab)}Na,b=1 defines the map. Let α = (a, b) be a composite index with a, b ∈

{1, . . . , N}, and rewrite (1) with m = N2 as L(X) =∑N

a,b=1 AabXBab with Aab, Bab ∈

CN×N . Aab and Bab can always be chosen to implement the map (2): the choice Aab = eab

leads to 〈ea|L(X)|ed〉 = L(X)ad =∑

b,cXbc (Bab)cd. The same matrix element of (2) is∑b,cXbc L(ebc)ad, so the choice (Bab)cd = L(ebc)ad reduces this to (2) as required. �

Definition 2 (Hermitian map). Let T : B(H,C)→ B(H,C) be a map on bounded linear

operators satisfying T (X)† = T (X†) for every X ∈ B(H,C). Then T is a Hermitian map

on B(H,C).

Hermitian maps have the property T (Her(H,C)) ⊆ Her(H,C), and they can be nonlinear.

Definition 3 (Positive map). Let φ : B(H,C)→ B(H,C) be a Hermitian map on bounded

linear operators satisfying φ(X � 0) � 0 for all X ∈ B(H,C), where · � 0 means it’s an

element of the PSD subset Her≥0(H,C). Then φ is a positive map on B(H,C).

We reserve the symbol φ for positive maps.

Proposition 1 (Linear positive map [61, 62]). Let φ : B(H,C)→ B(H,C) be a positive

linear map on finite-dimensional bounded linear operators. Then φ has a representation1

X 7→ φ(X) =m∑α=1

λαAαXA†α, Aα ∈ CN×N , tr(A†αAβ) = δαβ, λα ∈ R, m≤N2. (3)

Proof: Define a Choi operator C :=∑N

i,j=1 φ(eij) ⊗ eij in an expanded Hilbert space

HA⊗HB consisting of two copies of our system H, with dim(HA,B) = N. The matrix

elements of C in the product basis {|ea〉 ⊗ |eb〉}Na,b=1 are (〈ea| ⊗ 〈eb|)C (|ea′〉 ⊗ |eb′〉) =

〈ea|φ(ebb′)|ea′〉 = φ(ebb′)aa′ . Using the Hermitian property φ(X)† = φ(X†) we see that

C ∈ CN2×N2is Hermitian: (〈ea| ⊗ 〈eb|)C† (|ea′〉 ⊗ |eb′〉) =

(〈ea′ | ⊗ 〈eb′ |C |ea〉 ⊗ |eb〉

)∗=

φ(eb′b)∗a′a = φ(ebb′)aa′ = (〈ea| ⊗ 〈eb|)C (|ea′〉 ⊗ |eb′〉). It therefore has a spectral decompo-

sition C =∑N2

α=1 λα |Vα〉〈Vα|, λα ∈ R, where the eigenvectors {|Vα〉}N2

α=1 form a complete

orthonormal basis in HA⊗HB. Any |ψ〉 ∈ HA⊗HB can be obtained by the application of an

operator N12 A⊗ IN to |Ψ〉 := N−

12

∑Ni=1 |ei〉 ⊗ |ei〉, with Aij = 〈ei|A|ej〉 = (〈ei| ⊗ 〈ej|) |ψ〉.

1 The existence of the representation (3) does not actually require φ to be positive, only Hermitian.

6

Then |Vα〉〈Vα| = N (Aα ⊗ I)|Ψ〉〈Ψ|(A†α ⊗ I) where 〈ei|Aα|ej〉 = (〈ei| ⊗ 〈ej|) |Vα〉. Also note

that |Ψ〉〈Ψ| = N−1∑N

i,j=1 eij ⊗ eij, so

C = NN2∑α=1

λα (Aα ⊗ I)|Ψ〉〈Ψ|(A†α ⊗ I) =N∑

i,j=1

( N2∑α=1

λαAαeijA†α

)⊗ eij. (4)

Then φ(eij) =∑N2

α=1 λαAαeijA†α, and by linearity and completeness, φ(X) =

∑N2

α=1 λαAαXA†α,

as required. Furthermore, because the |Vα〉 are orthonormal, 〈Vα|Vβ〉 = tr(A†αAβ) = δαβ. �

Having established a general representation for positive linear maps, we give some non-

linear examples. The first is X 7→ φ(X) = (A†+CXB†)(A+BXC†) = [φ(X†)]†, A,B,C ∈

CN×N . On any PSD input, φ(X) = |A+BXC†|2 � 0, so φ is positive. Two examples of dis-

crete nonlinear positive maps are X 7→ φ±(X) = (tr(X)IN ±X)2 = [φ±(X†)]†, which reduce

to |tr(X)IN ± X|2 � 0 on PSD inputs. Another example is X 7→ φ(X) = |det(X)| · IN =

[φ(X†)]†, which (after normalization) maps every input to the infinite-temperature thermal

state. Permanents and determinants have also been used to construct nonlinear completely

positive trace-preserving (CPTP) channels [63].

Definition 4 (PTP channel [60]). Let Λ : Her≥01 (H,C)→ Her≥0

1 (H,C) be a positive map

satisfying tr(Λ(X)) = 1 for all X ∈ Her≥01 (H,C). Then Λ is a positive trace-preserving

(PTP) channel.

In this paper Λ always refers to a PTP channel. The PTP channels defined here are endo-

morphisms on the state space Her≥01 (H,C), and are only required to act properly on physical

states. They may be composed of maps defined on operators outside of Her≥01 (H,C) as well.

In this case, the behavior of the extended PTP channels on inputs outside of Her≥01 (H,C)

is not constrained by Definition 4.2

II. PTP MODELS

In this paper we investigate models of nonlinear quantum computation based on a cate-

gory of PTP channels that take the form of a nonlinear positive channel rescaled to conserve

trace [50, 51, 53, 61]. Although normalized PTP channels do not provide an exhaustive clas-

2 In particular, they might not be trace preserving in the usual sense of tr(Λ(X))=tr(X), because the trace

preservation in Definition 4 is really a trace fixing condition.

7

sification of nonlinear channels, they are sufficient to illustrate some of the different types

of effective nonlinearity that are available or might become available in the near future.

Definition 5 (Normalized PTP channel [50, 51, 53, 61]). Let φ : Her≥01 (H,C) →

Her≥01 (H,C) be a positive map satisfying tr[φ(X)] 6= 0 for all X ∈ Her≥0

1 (H,C). Then

the PTP map

Λφ : Her≥01 (H,C)→ Her≥0

1 (H,C) given by X 7→ Λφ(X) =φ(X)

tr[φ(X)](5)

is a normalized PTP channel.

Normalized PTP channels are common because they can be constructed from any positive

map φ satisfying a normalizability condition tr[φ(X)] 6= 0.3 From positivity, tr[φ(X)] ≥ 0.

Normalizability requires the more restrictive condition that tr[φ(X)] > 0. The condition

tr[φ(X)] > 0 is important because it ensures the positivity of Λφ. Normalized PTP channels

naturally fall into four classes, according to whether φ is linear (or not) and trace-preserving

(or not):

II A. Linear positive φ and tr[φ(X)] = 1 for all X ∈ Her≥01 (H,C). This is the class of

general linear positive channels, which includes linear CPTP channels.

II B. Linear positive φ and tr[φ(X)] 6= 1 for some X ∈ Her≥01 (H,C). These are called

nonlinear in normalization only (NINO) channels.

II C. Nonlinear positive φ and tr[φ(X)] = 1 for all X ∈ Her≥01 (H,C). This class includes

state-dependent CPTP channels.

II D. Nonlinear positive φ and tr[φ(X)] 6= 1 for some X ∈ Her≥01 (H,C). These are the

most general channels considered here. They support rich dynamics similar to that of

classical nonlinear systems.

These classes are discussed below in Secs. II A - II D.

3 To be precise, normalized PTP channels are only defined on a subset of physical states Her≥01 (H,C),

because their action on inputs with tr[φ(X)] = 0 is not specified. However this situation does not arise in

the cases considered here.

8

A. Linear PTP and CPTP channels

A standard form for linear PTP channels is provided in Proposition 1. While every

linear positive map can be put in the form (3), not every map of the form (3) is positive

(because the λα can be negative). The necessary conditions for (3) to represent a positive

(P) or completely positive (CP) map can be obtained as follows: Let S> = {α : λα >

0} 6= ∅ and S< = {α : λα < 0} be the index sets of positive and negative Choi eigenvalues,

and decompose φ into φ = φ> − φ<, where φ>(X) =∑

α∈S> λαAαXA†α and φ<(X) =∑

α∈S< |λα|AαXA†α are each manifestly positive. Upon rescaling, each can be put into the

form Φ(X) =∑

αAαXA†α, Aα ∈ CN×N . Maps of this form also satisfy the stronger condition

of complete positivity, meaning that they remain positive when combined with a second

Hilbert space HB, of any finite dimension, on which the identity acts: [Φ⊗ idB](X � 0) � 0.

Here X ∈ B(HA ⊗ HB,C). Each term in Φ clearly has the required property: 〈ψ| (Aα ⊗

I)X(A†α ⊗ I) |ψ〉 = 〈ψα|X |ψα〉 ≥ 0 for every |ψ〉 ∈ HA⊗HB, where |ψα〉 = (A†α ⊗ I) |ψ〉.

The condition for positivity is therefore φ>(X) � φ<(X) for every X � 0, whereas the

condition for complete positivity is φ<(X) = 0. An important CP map is the completely-

positive trace-preserving (CPTP) channel, which has a nonnegative Choi spectrum λα ≥ 0

and therefore an operator-sum representation [61, 62]

X 7→ Φ(X) =m∑α=1

AαXA†α,

m∑α=1

A†αAα = IN , Aα ∈ CN×N , m≤N2. (6)

This follows from Proposition 1 (but with different A′s as to enforce trace conservation

and absorb a factor of√λα). The integer m is the Choi rank. In this paper Φ always refers

to a linear CP (and usually TP) map.

There is a large body of work on linear P channels that are not CP, called non-CP maps

[64–67]. The distinction between CP and non-CP maps is the possibility of non-CP maps

generating negative states on entangled inputs, requiring their restriction to a subset of the

state space where this unphysical output is avoided. Although non-CP operations are well

known for their use in entanglement detection [68], the question of whether linear non-CP

channels could provide a computational advantage over linear CP channels appears to be

largely unexplored. However, non-CP communication channels in which the environment is

measured (and the results used to correct the channel) enable increased capacity [69–71].

9

In the presence of nonlinearity, we also find that the distinction between CP and non-CP is

significant, due to the larger space of nonunitary processes supported by non-CP channels.

B. NINO channels

Definition 6 (NINO channel). Let φ : Her≥01 (H,C) → Her≥0

1 (H,C) be a positive linear

map with tr[φ(X)] 6= 0 for every X ∈ Her≥01 (H,C), and tr[φ(X)] 6= 1 for one or more

X ∈ Her≥01 (H,C). Then the PTP map

Λφ : Her≥01 (H,C)→ Her≥0

1 (H,C) given by X 7→ Λφ(X) =φ(X)

tr[φ(X)](7)

is called a nonlinear-in-normalization only (NINO) channel.

We stress that the positive map φ in Definition 6 is linear, but not unitary φU(X) =

UXU †, U †U=IN (because tr[φU(X)] = tr(X) = 1 for all X). NINO channels inherit, from

linear maps, the powerful ability to characterize them through their action on a complete

basis (because φ has this property). Next we obtain a general representation for NINO

channels.

Proposition 2 (NINO representation). Let Λ be a NINO map of the form (7) with φ

linear. Then Λ has a representation

X 7→ Λ(X) =

m∑α=1

ζαAαXA†α

tr(FX), F :=

m∑α=1

ζαA†αAα 6= IN , Aα ∈ CN×N , ζα = ±1, m≤N2.

(8)

Proof: This follows from Proposition 1 after substituting λα = ζα|λα|, ζα = sign(λα),

and rescaling the A′s. �

Our main objective is to study NINO channels from a dynamical perspective, via master

equations. Some of the analysis will apply to the other classes as well. Let X ∈ Her≥01 (H,C)

be the state of a physical system which evolves continuously in time according to

X(t) 7→ X(t+ ∆t) = Λ∆t,t

(X(t)

), ∆t ≥ 0, Λ0,t = id for all t ∈ R. (9)

10

Here Λ∆t,t is a two-parameter family of PTP channels continuous in both t and ∆t. Because

we want to study the simplest instances that illustrate computational advantages, we make

several additional simplifying assumptions:

(i) Stationarity: Λ∆t,t = Λ∆t for all t ∈ R.

(ii) Semigroup: Λs ◦ Λt = Λs+t, 0 ≤ s� 1, 0 ≤ t� 1.

(iii) The nonlinearity can be turned off, recovering linear CPTP evolution.

(iv) N = 2.

From (9) we have that Λ∆t, defined in (i), is continuous and satisfies Λ0 = id. Stationarity

excludes time-dependent Hamiltonians that arise when a physical system is driven with

time-dependent fields (precisely what we want to do to run a device). While it is possible

to formulate the problem with time-dependent generators and obtain some of the results

in terms of time-ordered exponentials, we will not cover that case here. Instead we assume

that the nonlinearity can be turned on and off, and while in the off state the full toolkit

of linear quantum information processing can be applied. The additional assumptions (i)

and (ii) are sufficient to define, for any PTP channel, a Markovian master equation that

extends linear Markovian CPTP evolution by the Gorini-Kossakowski-Sudarshan-Lindblad

(GKSL) equation [72, 73]. Our approach follows a large body of work on nonlinear master

equations [42–53]. Especially relevant are the recent papers by Kowalski and Rembielinski

[50], Fernengel and Drossel [46], and Rembielinski and Caban [53]. The restriction to qubits

is not used in the derivation of the master equations, but is needed for their subsequent

analysis.

Before deriving a master equation for (8), we briefly consider the m = 1 case to illustrate

one of the differences between NINO and linear CPTP channels. Suppose A = etL is a

continuous time-dependent linear operator infinitesimally generated by some L ∈ B(H).

Decompose the generator into Hermitian and anti-Hermitian contributions L = L+ + L−,

with L± := (L ± L†)/2. Then X(0) 7→ X(t) = Λt(X(0)) = (etLX(0) etL†)/[tr[etL

†etLX(0)]]

and

dX

dt= [L−, X] + {L+, X} − 2 tr(L+X)X. (10)

11

Here {·, ·} is an anticommutator. Tracing gives ddt

tr(X) = 2 tr(L+X)−2 tr(L+X) tr(X) = 0

assuming tr(X) = 1, showing how the nonlinear term fixes the normalization. The equation

of motion (10) includes unitary evolution generated by a Hamiltonian H = iL−, together

with linear dissipation and amplification (if L+ has positive eigenvalues). Or we can say

that the dynamics is generated by a non-Hermitian Hamiltonian4 Hnon := iL = H + iL+.

NINO channels expand the utilization of linear maps by conserving the trace nonlinearly.

Now we obtain a master equation for general m. Continuous one-parameter NINO chan-

nels have the form

Λt(X) =

m∑α=1

ζαAα(t)X A†α(t)

tr(FtX), Ft :=

m∑α=1

ζαA†α(t)Aα(t) 6= IN , ζα = ±1, t ≥ 0. (11)

The operators Aα(t) ∈ CN×N are analytic functions of time for t ≥ 0, which can be classified

into two types, jump and nonjump, according to their t → 0 behavior. We assume that

k > 0 of the m operators have nonzero limits

limt→0

Aα(t) = A0α 6= 0, α ∈ {1, · · · , k} (nonjump), (12)

while the others vanish

limt→0

Aα(t) = 0, α ∈ {k+1, · · · ,m} (jump). (13)

The jump/nonjump terminology comes from the unraveled stochastic picture, where one

or more rare but disruptive “jump” operations are applied randomly to a simulated open

system, on top of a smooth background of unitary plus nonunitary evolution, with the

nonunitary component fixed to conserve trace. There is no restriction on the number of

jump operators; however no more than m2 are required. The only restriction on the number

of nonjump operators is that k > 0: At least one is required to obtain the desired t → 0

limit. Note that if Ft = IN , the nonlinear generator disappears and we recover linear GKSL

evolution [72, 73].

The constant matrices A0α ∈ CN×N in (12) are not arbitrary; the condition limt→0 Λt(X) =

X for all X ∈ Her≥01 (H,C) requires A0

α = zα IN , α ∈ {1, · · · , k}, where the zα ∈ C satisfy a

4 We distinguish between infinitesimal generators G and Hamiltonians iG. Thus, unitary evolution results

from Hermitian Hamiltonians but from anti-Hermitian generators.

12

“normalization” condition∑k

α=1 ζα|zα|2 = 1 (recall that ζα = ±1). To satisfy the semigroup

property it is sufficient to let

Aα(t) =

zα etLα for α ∈ {1, · · · , k},

Bα

√t for α ∈ {k+1, · · · ,m},

(14)

where Lα, Bα ∈ B(H,C). To see why, note that in the short-time limit

m∑α=1

ζαAα(t)XA†α(t) = X + t

[ k∑α=1

ζα|zα|2(LαX +XL†α

)+

m∑α>k

ζαBαXB†α

]+O(t2) (15)

and

Ft =m∑α=1

ζαA†α(t)Aα(t), (16)

= IN + t

[ k∑α=1

ζα|zα|2(Lα + L†α

)+

m∑α>k

ζαB†αBα

]+O(t2), (17)

= IN + tdF0

dt+O(t2). (18)

Then, if tr(X) = 1,

Λt(X) = X + t

[ k∑α=1

ζα|zα|2(LαX +XL†α

)+

m∑α>k

ζαBαXB†α − tr

(XdF0

dt

)X

]+O(t2) (19)

and

Λs(Λt(X)) = Λs+t(X) +O(s2) +O(st) +O(t2) for every X ∈ Her≥01 (H,C), (20)

as required for short times. The NINO evolution equation follows from (19):

dX

dt=

k∑α=1

ζα|zα|2(LαX +XL†α

)+

m∑α>k

ζαBαXB†α −X tr

(XdF0

dt

)

=k∑

α=1

ζα|zα|2(

[Lα−, X] + {Lα+, X})

+m∑α>k

ζαBαXB†α −X tr

(XdF0

dt

), (21)

where, in the second line, each linear operator Lα has been decomposed into Hermitian

and anti-Hermitian parts according to Lα = Lα+ + Lα−, with Lα± := (Lα ± L†α)/2. The

13

anti-Hermitian {Lα−}kα=1 each generate a unitary time evolution with Hamiltonian Hα =

iLα− = H†α, whereas the {Lα+}kα=1 and {Bα}mα>k generate nonunitary time evolution. The

brackets in (21) are commutators and anticommutators {A,B} = AB + BA. Note that

the parameters zα can be absorbed into rescaled generators Lα = |zα|2 Lα with no essential

change.

A principal difference between Markovian NINO and Markovian CPTP evolution is in

the form of the nonjump operators. In a CPTP channel, ζα = 1 and trace is conserved

through the requirement∑k

α=1 |zα|2Lα+ = −12

∑mα>k B

†αBα, which sets dF0/dt = 0 in (21).

In this case total Hermitian generator∑k

α=1 |zα|2Lα+ is always negative semidefinite, leading

to nonexpansive evolution and usually to a single stable fixed point. However in a NINO

channel the Lα+ are free parameters, and they can have positive eigenvalues. An example

of this distinction occurs when m = 1: Rank 1 CPTP channels are unitary and nondissi-

pative, whereas m = 1 NINO channels already support dissipation and amplification [see

(10)]. Therefore we can think of the NINO master equation (21) as a generalization of the

linear GKSL equation [72, 73] to support linear evolution by one or more non-Hermitian

Hamiltonians.

C. State-dependent CPTP channels

Next we discuss the class of normalized PTP channels (5) with nonlinear positive φ and

tr[φ(X)] = 1 for all X ∈ Her≥01 (H,C). These include the important subset of parametrically

nonlinear CPTP channels, which we call state-dependent CPTP channels.

Definition 7 (State-dependent CPTP). Let X ∈ B(H,C) and Aα(X) ∈ CN×N be a set

of X-dependent matrices satisfying

1. Aα(X†) = Aα(X),

2.m∑α=1

Aα(X)†Aα(X) = IN ,

for all X ∈ B(H,C) and any finite m. Then

X 7→ Λ(X) =m∑α=1

Aα(X)X Aα(X)† (22)

is a state-dependent CPTP channel.

14

Channels in this class have been investigated by many authors [2, 10–13, 28, 46, 48]. The

associated master equation is the state-dependent GKSL equation [72, 73]. Many early

proposals for nonlinear extensions of quantum mechanics, including the Weinberg model

[21], and unitary models based on a nonlinear Schrodinger equation, are in this class. A

rank 1 example is

X 7→ Λ(X) = U(X)X U(X)†, U(X) := ei tr(AX)B = U(X†), A,B ∈ Her(H,C). (23)

This map applies a generator B scaled by the mean 〈A〉 = tr(AX) of observable A. A

generalization of (23) to multiple nonlinear generators is U(X) = ei∑α tr(AαX)Bα , which

includes arbitrary state-dependent Hamiltonians and unitary mean field theories, including

the Gross-Pitaevskii equation for interacting bosons.

For a qubit in the Pauli basis, X = (I2 + r · σ)/2 ∈ Her≥01 (H,C), r = tr(Xσ) ∈ B1[0],

any Markovian PTP master equation can be put in the form5

dX

dt=σa

2

(dra

dt

),dra

dt= tr

(dX

dtσa)

= Gab(r) rb + Ca, G(r) ∈ R3×3, Ca ∈ R3, (24)

where we sum over repeated indices a, b ∈ (1, 2, 3). Here G(r) is a state-dependent gen-

erator, which can be decomposed into linear and nonlinear parts: Gab(r) = Lab + Nab(r).

If Nab(r) = 0, (24) describes a general affine transformation on X and r (strictly linear if

Ca = 0). Every G(r) can be decomposed into symmetric and antisymmetric components

G = G+ + G−, with G± := (G ± G>)/2, which have distinct actions on the Bloch vec-

tor length: ddt|r|2 = 2Gab(r) rarb = 2Gab

+ (r) rarb. Antisymmetric components G− conserve

Bloch vector length; they result from (possibly state-dependent) “unitary” transformations

X 7→ U(X)X U(X)†. Linear antisymmetric generators correspond to rigid rotations of the

Bloch ball and result from strictly linear unitary transformations on X. General symmetric

generators G+(r) can amplify some qubit states, increasing their Bloch vector, while de-

creasing others. Linear symmetric generators G+ resulting from entropy-increasing CPTP

channels have nonpositive G+ [see discussion following (21)] and cannot increase |r|. A

process that increases (decreases) |r| is called amplifying (dissiptive). Note that channel

characterization via changes in |r| is not specifically sensitive to nonlinearity. Next we con-

5 Let dX/dt= Y (X) = ξa(X)σa = ξa(r)σa where each ξa : R3 → R is a continuous function of r, which

can be decomposed as ξa(r) = 2Cα + 2Gab(r)rb = 2Cα + 2Labrb + 2Nab(r)rb, where 2Cα captures any

r-independent part of ξa(r).

15

sider a geometric characterization of the evolution: The divergence of the qubit velocity field

is

∇ · (dr/dt) = tr[G+(r)] + rb∂aGab(r), (25)

which has contributions from both symmetric and nonlinear generators, and can take either

sign. By contrast, the divergence is nonpositive in linear CPTP channels (because G+ � 0).

The vorticity

ω = ∇× (dr/dt), ωa = εabc∂b[Gcd(r)rd] = εabcGcb(r) + εabc[∂bG

cd(r)] rd (26)

also has linear and nonlinear contributions (ε is the Levi-Civita symbol). The εabcGcb(r) term

will contribute if G(r) ∈ R3×3 has an antisymmetric (|r|-conserving) part. The divergence

and vorticity characterize the velocity field, but don’t fully expose the computational benefits

of nonlinearity. This is because the velocity field describes how single states Xα follow

their streamlines, but does not directly convey the relative motion between evolving states.

For a more sensitive characterization we want to consider how pairs of states (Xα, Xβ)

transform under the channel. To further motivate this, consider a common setting for

quantum algorithms, where a subroutine accepts as input a sequence of quantum states

(X1, X2, X3, · · · ), then applies the same channel Λ to each in order to learn something about

those states or compute some function of those states. For example, we might know that

the states can only take values from a given set {Y1, Y2, · · · }, and we want to identify which.

Previous authors [1–3, 7, 13] have noted the intriguing computational power afforded by the

ability to increase the distinguishability between a pair of potential inputs Xα and Xβ, i.e., to

increase their trace distance ‖Xα−Xβ‖1, which is prohibited in linear CPTP channels. Let’s

examine this for a qubit in the Pauli basis: The differential of ‖X‖p := [tr(|X|p)]1p for any

square matrixX is d‖X‖p = ‖X‖1−pp tr(|X|p−1d|X|). Now letX = Xα−Xβ = 1

2(rα−rβ)·σ be

the difference between a pair of qubit states with Bloch vectors rα,β ∈ B1[0] and separation

‖Xα −Xβ‖p = 21p−1 |rα − rβ| measured in Schatten norm. Then

d

dt‖Xα −Xβ‖p = 2

1p−1 rα − rβ|rα − rβ|

·(drαdt− drβ

dt

)≤ 2

1p−1

∣∣∣∣drαdt − drβdt

∣∣∣∣ (27)

characterizes the expansivity of the channel: ddt‖Xα −Xβ‖p < 0 means that the channel is

16

strictly contractive on the pair, ddt‖Xα−Xβ‖p = 0 means it’s distance preserving on the pair,

and ddt‖Xα − Xβ‖p > 0 means it’s expansive. Expansivity allows for the distance between

two nearby states (Xα, Xβ) to increase. In the notation of (24) the rate of change of state

separation is

d

dt‖Xα −Xβ‖p = 2

1p−1 raα − raβ|rα − rβ|

[Gab(rα) rbα −Gab(rβ) rbβ

]. (28)

Expanding about the midpoint R = (rα + rβ)/2 gives, to second order in |rα − rβ|,

d

dt‖Xα −Xβ‖p = 2

1p−1 [Gab(R)+Kab(R)](raα−raβ)(rbα−rbβ)

|rα − rβ|, Kab(R) := Rc ∂bG

ac(R). (29)

We note the two distinct sources of expansivity in (29): Any antisymmetric part of G(R),

if present, doesn’t contribute to the expansivity, but positive eigenvalues in the symmetric

part do. This is an alternative expression of the same results we found above for d|r|2/dt,

and for the tr[G+(r)] term in the divergence. The second term contributes to expansivity if

the symmetric part of the matrix K ∈ R3×3 has positive eigenvalues.

In the remainder of this section we apply this geometric characterization to a state-

dependent CPTP channel with torsion [2, 13],

dra

dt= Gab(r) rb, Gab(r) = gzJabz , Jz =

0 −1 0

1 0 0

0 0 0

, g ∈ R. (30)

Here Jz is an SO(3) generator. G(r) ∈ R3×3 is antisymmetric and hence |r|-preserving. G(r)

generates z rotations with a rate that increases linearly with Bloch coordinate z, changing

direction for z < 0, a type of twist. The divergence (25) vanishes everywhere and the flow

is incompressible. The vorticity (26) is ω = (−x,−y, 2z)g. The z component ω3 describes

rigid body rotation within each plane of constant z, with a z-dependent frequency, while ω1

and ω2 reflect the associated shear. We can use (29) to discover expansive trajectories: In

the torsion model (30), the matrix Kab(R) defined in (29) is

K =g

2

0 0 −(yα + yβ)

0 0 (xα + xβ)

0 0 0

, K+ =g

4

0 0 −(yα + yβ)

0 0 (xα + xβ)

−(yα + yβ) (xα + xβ) 0

. (31)

17

K+ has eigenvalues 0 and ±(|g|/4)√

(xα + xβ)2 + (yα + yβ)2. For a pair of nearby states

Xα, Xβ, their difference rα−rβ is a short vector located at midpoint position R = (rα+rβ)/2.

Expansive trajectories occur when Kab(R)(raα−raβ)(rbα−rbβ) = ∂bGac(R)Rc(raα−raβ)(rbα−rbβ)

is positive. In the torsion model this condition simplifies to

g[Rx(yα − yβ)−Ry(xα − xβ)

](zα − zβ) > 0. (32)

Let rα = (12, ηy

2, ηz

2) and rβ = (1

2,−ηy

2,−ηz

2) be a pair of states with midpoint position

R = (12, 0, 0) along the positive x axis. The states are separated by ηy in the y direction and

ηz in the z direction. For nonzero ηy and ηz, the two states move in opposite directions and

separate at a rate ddt‖Xα −Xβ‖p = 2

1p−2g(ηyηz/

√η2y + η2

z).

D. General normalized PTP channels

Next we discuss channels with nonlinear positive φ and tr[φ(X)] 6= 1 for some X ∈

Her≥01 (H,C), the most general PTP channels considered here. This class combines the

nonunitary features of the NINO channels with the nonlinearity of state-dependent CPTP

channels. Suppose we want to add linear dissipation/amplification to the torsion model

(30) by adding a linear part G to the generator. What are the allowed values of G?

To answer this question, we use generators from the NINO master equation (21), namely

dX/dt = [L−, X] + {L+, X} + ζ2BXB†, where we have included one jump operator B and

one nonjump operator L.6 In the Pauli basis the first term in dX/dt leads to dra/dt = Gabrb,

with Gab = tr(σaL−σb − σbL−σ

a)/2, resulting in an antisymmetric contribution to G.

To see its connection with unitary dynamics, expand L− = −L†− in the Pauli basis as

L− = i(ξ0I + ξaσa), where ξ0, . . . , ξ3 ∈ R are real coordinates for L−. In this basis

Gab = 2εabcξc = −i tr(L−σc) εabc, a real but otherwise arbitrary linear combination of SO(3)

generators. Any linear antisymmetric G can be implemented by controlling these genera-

tors. Similarly, the {L+, X} term leads to a real symmetric Gab = tr(σaL+σb + σbL+σ

a)/2

plus an inhomogeneous part Ca = tr(σaL+). Expanding L+ = L†+ in the Pauli basis

as L+ = ξ0I + ξaσa, where ξ0, . . . , ξ3 ∈ R are again real, leads to a diagonal matrix

G = 2ξ0I3 = tr(L+)I3. And the ζ2BXB† term leads to Gab = ζ2 tr(σaBσbB†)/2 and

6 Here we assume that ζ1 =1; the normalization condition on the zα is then satisfied with z1 = 1.

18

Ca = ζ2 tr(σaBB†)/2 in the Pauli basis. Expanding B ∈ C2×2 as B = ξ0I + ξaσa, with

ξ0, . . . , ξ3 ∈ C complex coordinates for B, we have

Gab = ζ2

(|ξ0|2 − |ξ1|2 − |ξ2|2 − |ξ3|2

)δab + 2ζ2 Im(ξ∗0ξc)ε

abc + 2ζ2 Re(ξ∗aξb). (33)

The first term is diagonal. The second term is antisymmetric (both L− and B contribute to

unitary evolution if this term is nonzero). The third term is symmetric. Let ξ0 = 0; then

G = −ζ2 (|ξ1|2 + |ξ2|2 + |ξ3|2)I + 2ζ2 Re

ξ∗1

ξ∗2

ξ∗3

⊗(ξ1 ξ2 ξ3

) , (34)

where I is the identity. Consider now a pair of jump operators with the same ζ2 and

coordinates ξ = (1, 1, 0) and (0, 0, 1):

G(1,1,0) = ζ2

0 2 0

2 0 0

0 0 −2

, G(0,0,1) = ζ2

−1 0 0

0 −1 0

0 0 1

. (35)

Combining them gives

G(1,1,0) +G(0,0,1) = ζ2(2λ1 − I), λ1 =

0 1 0

1 0 0

0 0 0

, (36)

where λ1 is a Gell-Mann matrix. Similarly, G(1,0,1) + G(0,1,0) = ζ2(2λ4 − I) and G(0,1,1) +

G(1,0,0) = ζ2(2λ6 − I), where λ4 and λ6 are Gell-Mann matrices. By combining Hamiltonian

control with jump operator engineering, a set of linear generators G can be implemented.

19

III. FAULT-TOLERANT NONLINEAR STATE DISCRIMINATION

In the remainder of the paper we consider an extension of the qubit torsion channel (30)

that includes linear dissipation and amplification. Using the techniques of Sec II D, jump

operators are chosen such that

dX

dt=σa

2

(dra

dt

),dra

dt= tr

(dX

dtσa)

= Gab(r) rb = (mλ4 − γI + gzJz)abrb, (37)

where I is the 3×3 identity,

λ4 =

0 0 1

0 0 0

1 0 0

, and Jz =

0 −1 0

1 0 0

0 0 0

. (38)

Here λ4 is an SU(3) generator, Jz is an SO(3) generator, and we sum over repeated indices

a, b ∈ (1, 2, 3). The dimensionless model parameters m, γ, and g are real variables of either

sign. The fixed-point equations are

dx

dt= mz − γx− gyz = 0, (39)

dy

dt= −γy + gxz = 0, (40)

dz

dt= mx− γz = 0. (41)

The origin is always a fixed point, rfp0 = (0, 0, 0), although not always stable. Assuming

r 6= (0, 0, 0), γ 6= 0, and eliminating z, the fixed point equations are

m2 − γ2 = gmy, (42)

γ2y = gmx2. (43)

If g = 0, any fixed points must be confined to the y = 0 plane. There are no additional

fixed points unless γ = ±m, in which case there is a set of fixed points rfp,g=0z=±x on the line

rz=±x = {(x, 0,±x) : x ∈ R}. The settings γ = ±m are singular lines in the parameter

space of the model. Manipulating these singularities in the presence of nonlinearity is the

key to engineering useful information processing.

20

When g > 0, any fixed points must be confined to the plane my = (m2− γ2)/g. However

(43) requires y to have the same sign as that of m. Therefore my > 0, which is only possible

when m2 > γ2. Therefore, when m2 < γ2, the only fixed point is rfp0 , and this fixed point is

stable for all m2 < γ2. If instead the condition m2 > γ2 is satisfied, and g > 0, there is a

pair of stable fixed points at

rfp± =

(± |γ|

g

√δ,

m

gδ, ± sign(γ)

m

g

√δ

), δ :=

m2 − γ2

m2∈ (0,∞]. (44)

For these fixed points to be contained within the Bloch ball requires |g| > gmin, where

gmin =√

(γ2 +m2)δ +m2δ2. The dynamics between fixed points rfp− , r

fp0 , and rfp

+ can be

understood as follows: When g = 0 we have

dx

dt= mz − γx, (45)

dy

dt= −γy, (46)

dz

dt= mx− γz. (47)

Note that the y motion is decoupled from x and z, and that it is always stable for γ > 0.

Furthermore, the linearized model has an additional symmetry which becomes explicit after

changing variables to ξ± = (z ± x)/2:

dξ+

dt= (m− γ)ξ+, (48)

dξ−dt

= −(m+ γ)ξ−. (49)

The ξ+ and ξ− variables are also decoupled. ξ+ is the coordinate along the line z = x

mentioned above, and ξ− is the coordinate along the perpendicular line z = −x. Motion in

the ξ+ direction is stable for m < γ; in this case each point on the line z = x flows to the

fixed point rfp0 at the origin. However the ξ+ motion becomes unstable when m > γ. In this

regime rfp0 is unstable, and each point on the line z = x (other than z=x=0) flows outward

to infinity. We can interpret this unstable case as having two stable fixed points at (∞, 0,∞)

and (−∞, 0,−∞), at the ends of the line z = x. By contrast, close to the singularity at

m = γ, the perpendicular ξ− motion is stable unless m and γ are both negative. In this

picture, the most important effect of the nonlinearity is to move the two stable fixed points

21

FIG. 1. Illustration of the dynamics in the neighborhood of the origin for m, γ ≥ 0. When m < γ,

two unstable fixed points at infinity (red) feed the stable fixed point rfp0 at the origin (black).

However, if m > γ, rfp0 is unstable (red). The ξ− axis (green) is the separatrix. rfp

0 feeds two new

stable fixed points rfp+,− (black). At the critical point m = γ, all points on the ξ+ axis (blue dots)

are fixed points.

at infinity to the finite positions (44). This is illustrated in Fig. 1. In Fig. 2 we plot the

trajectories for a cloud of randomly chosen initial states (red dots) within the Bloch ball

B1[0], close to the bifurcation but in the m2 < γ2 phase. In Fig. 3 we show the same plot in

the m2 > γ2 phase. These simulations further support the picture described above.

The ξ+ dynamics near the unstable fixed point rfp0 can be used to achieve robust state

discrimination with the exponential speedup supported by expansive nonlinear channels [1–

3, 7, 13]. Points very close to rfp+ have |r| � 1, so the nonlinearity can be neglected there.

Equations (48) and (49) then apply to the dynamics near rfp0 even when g 6= 0. Consider

the plane passing through the origin and perpendicular to the ξ+ axis. The velocity field

smoothly changes sign across this plane; i.e., it is a separatrix between basins of attraction

for rfp+ and rfp

− . Suppose that a qubit is prepared in a state from the set {Xα, Xβ}, with

Xα and Xβ close in trace distance ε = ‖Xα −Xβ‖1 but on opposite sides of the separatrix.

Then we implement a gate by turning on the nonlinearity for a time t = O(1/g), after which

Xα and Xβ flow to different fixed points. After this evolution, the nonlinearity is turned off

and the qubit is measured. This nonlinear gate would lead to an exponential speedup if the

initial separation is exponentially small: ε = 2−k [2, 7, 13].

Positivity of the dissipative torsion channel requires that the Bloch vector r remain in the

Bloch ball |r| ≤ 1. However the master equation (37) does not itself enforce this condition.

22

FIG. 2. Simulation solutions of the torsion channel (37) with γ = 1, m = 0.9, and g = 1, showing

attraction to rfp0 . The x, y, and z axes are Bloch vector coordinates. Red dots indicate random

initial conditions. The Bloch sphere is outlined in yellow.

FIG. 3. Simulation solutions to (37) with γ = 1, m = 1.1, g = 1, and δ = 0.2, showing attraction

to rfp± .

Therefore the positivity condition must be implemented dynamically through control of the

qubit Hamiltonian, and trajectories leaving the Bloch ball are regarded as unphysical.

IV. DISCUSSION

Although this gate is appealing, it has limitations:

(i) First, finite experimental resolution and control will limit the smallest values of ε

23

achievable in practice. If the inputs to the discriminator are the outputs of a preceding

process, they will also come with errors.

(ii) Second, the fixed points rfp+,− are not perfectly distinguishible. Although there is

considerable flexibility in choosing their location, in practice they need to be well

within the Bloch ball to ensure positivity. If the rfp+,− are too close to the surface

|r| = 1, trajectories approaching them may lead to unphysical solutions leaving the

Bloch ball.

(iii) And third, there will likely be errors associated with the effective model itself.

In conclusion, we have discussed three classes of nonlinear PTP channels and explored

their computational power. Engineering both nonlinearity and dissipation allows one to

implement rich dynamics similar to that of classical nonlinear systems. We identified a

bifurcation to a phase where the Bloch ball separates into two basins of attraction, which

can be used to implement fast quantum state discrimination [2, 7, 13] with intrinsic fault-

tolerance.

ACKNOWLEDGEMENTS

It is a pleasure to thank Andrew Childs for correspondence.

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